Spectral convergence of the Dirac operator on typical hyperbolic surfaces of high genus

In this article, we study the Dirac spectrum of typical hyperbolic surfaces of finite area, equipped with a nontrivial spin structure (so that the Dirac spectrum is discrete). For random Weil-Petersson surfaces of large genus $g$ with $o(\sqrt{g})$ cusps, we prove convergence of the spectral density to the spectral density of the hyperbolic plane, with quantitative error estimates. This result implies upper bounds on spectral counting functions and multiplicities, as well as a uniform Weyl law, true for typical hyperbolic surfaces equipped with any nontrivial spin structure.

1. Introduction 1.1.Setting and motivation.The objective of this article is to provide information on the Dirac spectrum of typical hyperbolic surfaces of genus g with k cusps, where g, k are non-negative integers such that 2g −2+k > 0. In order to do so, we equip the moduli space of hyperbolic surfaces of signature (g, k) with the Weil-Petersson probability measure P g,k .This is a natural model to study typical hyperbolic surfaces, as illustrated by the rich literature that has developed in the last few years [1,6,7,8,10,11,13,20,19].By typical, we mean that we wish to prove properties true with probability going to one in a certain asymptotic regime.
An example of interesting regime is the large-scale regime, i.e. the situation when g and/or k go to infinity.Indeed, by the Gauss-Bonnet formula, the area of any hyperbolic surface of signature (g, k) is 2π(2g − 2 + k).Since g and k are two independent parameters, we can a priori expect typical surfaces of large genus or large number of cusps to exhibit different geometric and spectral properties.This has been confirmed by the recent complementary works of Hide [7] and Shen-Wu [15], which prove very different behaviours for the Laplacian spectrum depending on whether k ≪ √ g or k ≫ √ g.
In this article, we fix a sequence of non-negative integers (k(g)) g≥2 such that k(g) = o( √ g), i.e. k(g)/ √ g → 0 as g → +∞.This setting is the genus-dominated regime, and we leave the cusp-dominated regime k(g) ≫ √ g to further work.Under this hypothesis, the second author [12] and Le Masson-Sahlsten [9] have proven that there exists a set A g,k(g) of "good hyperbolic surfaces" of signature (g, k(g)) such that: • the Weil-Petersson probability of A g,k(g) goes to one as g → +∞; • the systole of any X ∈ A g,k(g) is bounded below by g −1/24 log(g); • elements X ∈ A g,k(g) are close to the hyperbolic plane in the sense of Benjamini-Schramm, and more precisely, the proportion of points on X of injectivity radius smaller than 1  6 log(g) is at most g −1/3 .We prove upper bounds and asymptotics for the Dirac spectrum of hyperbolic surfaces in A g,k(g) , which we shall now present.
1.2.Spectral convergence of the Dirac operator.For a hyperbolic surface X of signature (g, k), a spin structure ε on X, we denote as D the Dirac operator on (X, ε).
While the Laplacian spectrum of a hyperbolic surface with cusps always contains essential spectrum, equal to [1/4, +∞), for any X, one can pick ε so that the spectrum of the Dirac operator D is discrete [2].We call such spin structures nontrivial.In that case, for 0 ≤ a ≤ b, we denote as N |D| (X,ε) (a, b) the number of eigenvalues of the absolute value |D| of the Dirac operator D, once all rigid multiplicities are removed (see Section 2.2.2).
Our main result is the spectral convergence of the Dirac operator on (X, ε) to the Dirac operator on the hyperbolic plane H, true for any typical hyperbolic surface X of high genus g with o( √ g) cusps, and any nontrivial spin structure ε on X.
There exists a constant C > 0 such that, for any 0 ≤ a ≤ b, any g ≥ 2, any X ∈ A g,k(g) , and any nontrivial spin structure ε on X, we have where the remainder satisfies: A similar statement holds for the Laplacian spectrum, by work of the first author [13] in the compact case and Le Masson-Sahlsten [9] when k(g) ≪ √ g.These results have further been extended to twisted Laplacians by Gong [6] very recently.The proof is similar to the proof in [13], replacing the Selberg trace formula by a Dirac version from the second author [16].
Note that the support of the limiting measure is [0, +∞), because the spectrum of |D| on H is [0, +∞).This contrasts with the (twisted) Laplacian setting, where the limiting ,+∞) (λ) dλ, supported on [1/4, +∞).A remarkable aspect of Theorem 1 is that the limit we obtain is independent of the nontrivial spin structure ε.The reason for that is that the probabilistic assumption we make, and more precisely the Benjamini-Schramm hypothesis, makes the geometric term of trace formulae subdominant, i.e. the spectra of X converge to the spectra of H, regardless of the precise geometry and spin structure on X.
1.3.Upper bounds and pathological surfaces.In the process of proving Theorem 1, we prove the following upper bound on the Dirac spectrum of typical hyperbolic surfaces.Throughout this article, when we write A = O(B), we mean that there exists a constant C > 0 such that, for any choice of parameters, |A| ≤ C B. We precise that the constant is allowed to depend on our choice of a fixed sequence (k(g)) g≥2 .If the constant depends on a parameter p, e.g. the genus g, we rather write A = O p (B).
Building on results of Bär on pinched surfaces [2], we prove that such a bound cannot be obtained for every spin hyperbolic surface, because there exists "pathological" examples for which the Dirac operator is discrete with arbitrarily many eigenvalues close to 0. Proposition 3. Let (g, k) be integers such that 2g − 2 + k > 0 and g ≥ 1.For any N, any η > 0, there exists a hyperbolic surface X of signature (g, k) and a nontrivial spin structure ε on X such that N |D| (X,ε) (0, η) ≥ N.This is another interesting difference between Laplacian and Dirac spectra.Indeed, the Laplacian spectrum restricted to [0, 1/4] is discrete, and the number of eigenvalues under 1/4 is at most 2g − 2 + k by work of Otal-Rosas [14].This is a topological bound, in the sense that it only depends on the topology of the hyperbolic surface.Proposition 3 proves that such a bound cannot exist in the Dirac setting, while Proposition 2 provides one true for any typical hyperbolic surface.
1.4.Applications.We deduce from Theorem 1 a uniform version of the Weyl law for Dirac operators on typical hyperbolic surfaces (uniform in the sense that the rate of convergence is independent of the surface X ∈ A g,k(g) and the nontrivial spin structure ε).
Corollary 4. For any g ≥ 2, any X ∈ A g,k(g) , any nontrivial spin structure ε on X, The implied constant above only depends on the genus g, in a way that can be made explicit using Theorem 1.
Taking a shrinking interval of size 1/ log(g) above λ, we deduce from Proposition 2 the following topological bound on the multiplicity mult (X,ε) (λ) of any Dirac eigenvalue λ.
1.5.Acknowledgements.The authors would like to thank Sergiu Moroianu for valuable discussion and comments.This research was funded by the EPSRC grant EP/W007010/1 "Spectral statistics for random hyperbolic surfaces".The second author was partially supported from the project PN-III-P4-ID-PCE-2020-0794, financed by UEFISCDI.

Preliminaries
2.1.Spin structures and the Dirac operator.In this subsection we briefly describe spin structures and introduce the Dirac operator.For more details, we refer the reader to [4] and [5].
Let n be an even integer, and let us denote by Cl n the Clifford algebra associated to R n with the standard scalar product.The subgroup Spin(n) ⊂ Cl n consists of all even products of unit vectors.It can be shown that Spin(n) is a connected, two-sheeted covering of SO(n), the group of n × n matrices of determinant 1.Consider {e 1 , ..., e n } the standard basis in R n , and denote by J the standard almost complex structure.The representation: where is obtained form the C-bilinear extension of the standard scalar product, acts on One can check that this representation extends to the complexified Clifford algebra Consider X a n-dimensional oriented manifold with a Riemannian metric.A spin structure on X is a principal Spin(n) bundle P Spin(n) X covering P SO(n) X, the principal bundle of oriented orthonormal frames, with two sheets.Moreover, this covering must be compatible with the group covering Spin(n) −→ SO(n).Once we have a fixed spin structure, we can define the spinor bundle as the associated vector bundle S := P Spin(n) X × cl Σ n .The Dirac operator is defined as follows: where ∇ is the connection on S induced by the Levi-Civita connection on X.One can see that D is an elliptic, self-adjoint differential operator of order 1.
It is known that orientable, complete hyperbolic surfaces of finite area always admit spin structures.From now on, we restrict our attention to this type of surfaces.Let X = Γ \ H, where Γ is a subgroup of PSL 2 (R) without elliptic elements for which the associated surface is of finite area.We denote as π the standard projection π : SL 2 (R) −→ PSL 2 (R).We can reinterpret a spin structure on X as a left splitting in the following short exact sequence: i.e. a morphism χ : Γ −→ {±1} for which χ • ι = id {±1} , where ι is the natural inclusion.
We define ε : Γ −→ {±1} by setting ε(γ) := χ(γ), where γ ∈ SL 2 (R) is the unique lift with positive trace of γ ∈ PSL 2 (R).This function is a class function, i.e. constant along conjugacy classes.More details about the class function associated with a spin structure can be found in [16,Section 2].Further detail is also provided on the identifications between the frame bundle P SO(2) H and the group of isometries PSL 2 (R), and between the spin bundle P Spin(2) H and the group SL 2 (R).If we consider ρ, a right splitting in the above short exact sequence (which is uniquely determined by the left splitting χ), we can define the action of a γ ∈ Γ on P Spin(2) H by left multiplication with ρ(γ).It can be easily seen that this action descends to the spinor bundle S.

2.2.
The spectrum of Dirac operators.The aim of this article is to study the spectrum of the Dirac operator acting on a typical hyperbolic surface X of finite area.

Cusps and nontrivial spin structures.
The spectrum of the Dirac operator is always discrete when X is compact.Remarkably, when X admits some cusps, the spectrum can either be discrete or the real line R, depending on the spin structure.More precisely, Bär showed in [2, Theorem 1] that the spectrum of the Dirac operator is discrete if and only if the spin structure is nontrivial along each cusp of X.It is shown in [16,Lemma 8] that this is equivalent to assuming that ε(γ) = −1 for any primitive parabolic element γ ∈ Γ.
Note that Bär further proved in [2, Corollary 2] that any finite area hyperbolic surface admits at least one spin structure such that the spectrum is discrete.

Multiplicities and counting functions.
As explained in [3, Section 4], the spectrum of the Dirac operator admits two rigid sources of multiplicity: the chiral symmetry and the time-reversal symmetry.It follows that the spectrum of D is symmetric about 0 (i.e. if λ ∈ R is an eigenvalue then −λ is an eigenvalue), and every eigenvalue has even multiplicity.We conclude that the multiplicity of every eigenvalue of |D| is a multiple of 4.
In order to avoid counting every eigenvalue exactly four times, we shall study the reduced spectrum (λ j ) j≥0 , where we let λ j := Λ 4j for (Λ j ) j≥0 the ordered spectrum of |D| (with multiplicities).We then define, for 0 ≤ a ≤ b, the counting function

Pathological examples of Dirac spectra.
Let us now prove Proposition 3, which claims that there is no topological bound on the number of eigenvalues in [0, η], provided g > 0.
The proof relies on the two following results, proven by Bär in [2].
Lemma 6.Let X be a finite area hyperbolic surface.For any simple non-separating closed geodesic γ on X, there exists two nontrivial spin structures ε ± on X such that ε ± (γ) = ±1.
We precise that, in the previous statement, the geodesic γ is simple if it has no selfintersection, and non-separating if the surface X \ γ obtained by cutting X along γ is connected.In other words, this lemma tells us that, if γ is non-separating, then we can pick the value of a nontrivial spin structure at γ freely.
Proof.This is a direct consequence of the discussion in [2, page 481].
Lemma 7. Let X be a finite area hyperbolic surface equipped with a nontrivial spin structure ε.Let γ be a simple non-separating geodesic on X such that ε(γ) = +1.Let (X n ) n≥1 be a sequence of finite area hyperbolic surfaces obtained from X by pinching the geodesic γ so that its length goes to 0 as n → ∞.Then, for any η > 0, The pinching procedure mentioned above is a classic way to construct pathological examples in hyperbolic geometry, and described in more detail in [2, Section 1].Lemma 7 quantifies how the Dirac spectrum of X n converges to the Dirac spectrum of the limit X ∞ of (X n ) n as n → ∞.There is an accumulation process because, on top of the nontrivial cusps of X, X ∞ has two new cusps with a trivial spin structure (coming from the pinched geodesic), and hence the Dirac spectrum of (X ∞ , ε) is R.
Proof.This is a trivial adaptation of [2, Theorem 2] when X is a surface of finite area equipped with a nontrivial spin structure, rather than a closed surface.No changes are required in the proof.
We are now ready to prove Proposition 3.
Proof.Let X be an arbitrary hyperbolic surface of signature (g, k).The genus of X is nonzero, and hence there exists a simple non-separating geodesic γ on X.By Lemma 6, since γ is non-separating, there exists a nontrivial spin structure ε on X such that ε(γ) = +1.Then, we define a sequence of metrics (X n ) n≥1 as in Lemma 7 by pinching the geodesic γ so that its length goes to 0 as n → +∞.By Lemma 6, since ε(γ) = +1, for any fixed η > 0. In particular we can pick a n such that N |D| (Xn,ε) (0, η) ≥ N.Then, X n satisfies our claim.

The Selberg trace formula for Dirac operators.
Our main tool to study the counting function N |D| (X,ε) (a, b) is the Selberg trace formula for the Dirac operator on compact hyperbolic surfaces, developed by Bolte and Stiepan in [3] and generalised to hyperbolic surfaces of finite area by the second author [16,Theorem 13].This formula relates the Dirac spectrum of a finite area hyperbolic surface X to its length spectrum, i.e. the list of the lengths of all closed geodesics on X, under the condition that the spin structure is nontrivial.In this article, following the line of [13], we will use the following pretrace formula, adapted from [16, formula (10)].
Theorem 8. Consider X = Γ\H a hyperbolic surface with k cusps, equipped with a nontrivial spin structure ε.Let (λ j ) j∈N denote the reduced spectrum of |D|.Then, for any admissible test function h, where the sum is taken after all hyperbolic elements in Γ and: • the set F is a fundamental domain of X = Γ\H; • τ z →w = −i z−w |z−w| is the parallel transport of spinors from z to w with respect to ∇; • the kernel K can be expressed as: where ȟ is the inverse Fourier transform of h, i.e.
Proof.In [16, formula (10)], the second author proved that, under the hypotheses of the theorem, for any , and ȟ(x holds with the kernel K(r) := φ(4 sinh 2 (r/2)).As a consequence, in order to conclude, all that we have to do is to associate a function φ to our test function h, and hence express the kernel K in terms of h.To do so, we shall consider the following operators A, B acting on the set S([0, ∞]) of Schwartz functions on [0, ∞]: By direct computations, one can easily see that B • A = 1.Indeed: Writing the identity in such a way allows us to compute the kernel in terms of ȟ.On the one hand, by the expression of ȟ in equation ( 3), we get that: thus, it follows that: On the other hand, by definition of K and w, since 4 sinh 2 ( r 2 ) + 4 = 2 cosh( r 2 ), We then perform the change of variable 4 sinh 2 ( ρ 2 ) = 4 sinh 2 ( r 2 ) + y 2 and obtain the claimed expression thanks to (4) and the fact that 2.3.Random hyperbolic surfaces.In this article, we study the properties of random hyperbolic surfaces sampled with the Weil-Petersson probability measure.Let us provide the key elements that are necessary for the reading of this article -thorough presentations of this probabilistic model are provided in [12,18].Let g, k be integers such that 2g − 2 + k > 0. Our sample space is the moduli space This space is an orbifold of dimension 6g − 6 + 2k.Weil introduced in [17] a natural symplectic structure on M g,k , called the Weil-Petersson form.It induces a volume form of finite volume, which can be renormalised to obtain a probability measure P g,k on the moduli space M g,k .
Our objective is to describe "typical behaviour", i.e. we will focus on proving properties true with probability going to one in a certain asymptotic regime.More precisely, for our fixed sequence (k(g)) g≥2 , we will say a property is true with high probability in the large genus limit if lim g→+∞ P g,k(g) (X ∈ M g,k(g) satisfies the property) = 1.
The following result states two key geometric properties true with high probability which we will use in this article.
Theorem 9. Let (k(g)) g≥2 be a sequence of non-negative integers such that k(g) = o( √ g) as g → +∞.Then, for all g ≥ 2, there exists a subset A g,k(g) of the moduli space M g,k(g) of probability 1 − O(log(g)g −1/12 ) such that any surface X ∈ A g,k(g) satisfies the following.
• If X − (L) is the L-thin part of X, i.e. the set of points in X with radius of injectivity shorter than L, then • The systole of X (i.e. its shortest closed geodesic) is longer than g

Plan of the proof and first estimates
In this section, we set up some notations in order to prove Theorem 1, following the lines of [13], and prove first easy estimates.
3.1.The family of test functions.A key step of the proof is to construct a family of test functions such that the spectral side of the Selberg trace formula is a good approximation of the counting number N |D| (X,ε) (a, b), for 0 ≤ a ≤ b.Our choice of test function is a straightforward adaptation to the choice made in [13, Section 4].
For t > 0, a parameter which will grow like √ log g, consider the family of test functions h t : C −→ C defined by the convolution where ½ [a,b] is the indicator of the segment [a, b] and v t (x) := t √ π exp (−t 2 x 2 ) is the Gaussian of mean 0 and variance 1 t .One can easily see that h t is holomorphic.Since it is not even, we will rather apply Proposition 8 to the function H t (λ) := h t (λ) + h t (−λ).
Let us present elementary properties of the functions h t and g t := Ȟt proven in [13], which will be useful to the proof of Theorem 1.
(1) The function H t is admissible.
3.2.Plan of the proof.In order to prove our main result, we apply Theorem 8 to the family of functions H t , of kernels K t , and obtain that for any hyperbolic surface X of signature (g, k), ( The left hand side of this formula is an approximation of the ratio N |D| (X,ε) (a, b)/ Area(X), which we wish to estimate.Thus, we shall study the right hand side, term by term.
• In Section 3.3, we bound the difference between the integral term and the integral that appears in Theorem 1. • Then, in Section 3.4, we prove an easy bound on the cuspidal term g t (0).
• Section 4 is dedicated to bounding the kernel term, which is the most difficult part of the analysis of the trace formula, where the probabilistic assumption on X ∈ A g,k(g) is necessary.We then conclude to the proof of Theorem 1 in Section 5, where we compare the left hand side of (5) with the rescaled number of |D|-eigenvalues between a and b.

Asymptotic of the integral term.
Let us prove the following result, which bounds the difference between the integral I(t, a, b) and integral appearing in our claim.Proposition 11.For any t > 0, Proof.We start by rewriting I(t, a, b) in a more convenient form.First we use the parity of H t , then we write H t (λ) = h t (λ) + h t (−λ), to obtain If a + 1 t > b − 1 t then the interval in the middle is omitted from the union.The integral splits accordingly into five parts, denoted I j , for 1 ≤ j ≤ 5: We start with I 1 , I 3 and I 5 .Throughout the computations, we will use the fact that: Equation 7 and Lemma 10.(3) for λ < a together imply by the change of variable x = t(a − λ) ∈ (1, +∞).Then, With similar approximations one can also prove that: For I 2 and I 4 we rather use the loose bound h t (λ) − 1[a,b] (λ) ≤ 1, which yields In the same way we also get I 4 ≤ 2b+2 t + 2 t 2 .Combining those bounds and using a ≤ b, we obtain which allows us to conclude using equation ( 6) and 5/(4π) < 1/2.

Bond of the cusps contribution.
Let us now prove the following.
Proposition 12.For any X of signature (g, k) and any t > 0, Remark 13.We note that, in Theorem 1, we are placed in the regime k = k(g) = o( √ g).
Then, Proposition 12 implies Proof of Proposition 12.By definition, the cusp term is by the Gauss-Bonnet theorem.We therefore simply have to estimate Similarly we have that R h t (−λ) dλ = b − a.Hence, g t (0) = (b − a)/π, which leads to the claim.

Bound of the kernel term
In what follows we show a bound on the kernel term of the trace formula, where the summation runs over hyperbolic elements in the group Γ which are not the identity, and F is a fundamental domain of X = Γ\H.The steps of the kernel bound are as follows.
• First, in Section 4.1, we prove an upper bound on the values K t of the kernel appearing in R K .• We prove a classic counting bound on hyperbolic elements of Γ in Section 4.2, in order to deal with the summation.• We then cut the fundamental domain F in a thick and thin part, F ± (L), in Section 4.3.We bound the integrals over these two sets separately.• Finally, in Section 4.4, we conclude to a quantitative probabilistic bound on R K , using the probabilistic assumption from Theorem 9, and in particular the Benjamini-Schramm hypothesis.
4.1.Kernel estimate.Let us prove the following bound on K t .

4.2.
Bound on the number of hyperbolic elements.We shall use the following classic bound in order to control the summations over hyperbolic elements of Γ.
Lemma 15.Let r ≤ 2 be a positive number and let X = Γ\H be a hyperbolic surface whose systole is larger than 2r.Then, for any j > 0, any z ∈ H, Proof.Choose z ∈ H a point.The family of disks centred at γz and of radius r/2, for hyperbolic elements γ ∈ Γ, are disjoint.By comparing areas, the number of hyperbolic elements γ for which d(z, γz) ≤ j must be smaller than: 4.3.Thin-thick decomposition of the fundamental domain.For a positive real number L, we decompose the fundamental domain F of X = Γ\H as a disjoint union of two sets F − (L) and F + (L), the points of F with injectivity radius smaller than L and larger than L respectively.Splitting the integral in the sum R K into two integrals, on those two sets, we can rewrite our sum as: ).We shall start by bounding the contribution given by integration on F + (L), using the fact that all points on F + (L) have an injectivity radius larger than L. Lemma 16.Let t > 0 and 0 < r ≤ 2. Suppose that X = Γ \ H is a hyperbolic surface whose systole is larger than 2r.If L is a real number such that L ≥ 8t 2 , then: Proof.By definition of F + (L), for z ∈ F + (L), the sum defining R + K contains no elements γ such that d(z, γz) < L. Thus, we can write: When bounding the quantity above by the triangular inequality, we note that |ε(γ)| = |τ z →γ −1 z | = 1, which allows to safely ignore these terms.Moreover, notice that the distance between z and γz is always larger than r, the injectivity radius.Thus, Proposition 14 together with Lemma 15 imply: First, we note that Area(F + (L)) ≤ Area(X).We then observe that, provided L ≥ 8t 2 , we have j ≤ j 2 8t 2 and hence by comparison of the sum with an integral, which is bounded by 2 exp(−L) as soon as L ≥ 8t 2 , thus implying our claim.
We now bound the contribution of F − (L).
Lemma 17.With the notations of Lemma 16, We observe that, due to the fact that the injectivity radius on F − (L) is not bounded below by L ≫ 1, we do not obtain an exponential decay like e −L in Lemma 16.However, the ratio Area(F − (L))/ Area(X) will decay under the Benjamini-Schramm hypothesis.
Proof.As before, we combine Proposition 14 together with Lemma 15 to obtain: To deal with the last sum, we split it at ⌊8t 2 ⌋ + 1. Proceeding as in Lemma 16, we deduce: For remaining indices we just bound naively: which leads to the claim.
4.4.Probabilistic kernel estimate.The last step of this section is to use our probabilistic hypotheses, presented in Section 2.3, to bound the kernel term.We prove the following.
Proposition 18.Let 0 ≤ a ≤ b, g ≥ 2, and set t := On the one hand, we observe that the first and the third term on the right hand side of (12) can easily be bounded above using Proposition 2: On the other hand, we can proceed as in the proof of Proposition 2 to bound the second term of the right hand side, except more finely this time.More precisely, we write again and then use Lemma 19 to obtain a constant C > 0 such that We now observe that, for all 0 ≤ x ≤ 1/2, (1 − x) −1 ≤ 1 + 2x.We apply this inequality to x = e −t 2 η 2 /( √ πtη) ≤ 1/2 (thanks to the fact that tη ≥ 1) and get: for a constant C ′ .We can then conclude using the decomposition (12), the bounds ( 17) and ( 13) with our value of η specified in (16).
Proof of the lower bound of Theorem 1.Since 0 ≤ h t ≤ 1 everywhere one has: which is what we need to conclude.

− 1 24 √
log g.Proof.The first point was proven by the first author in [12, Corollary 4.4], and the second by Le Masson and Sahlsten in [9, Lemma A.1].